Question

Let X₁, X₂,...X₁₈ be eighteen observation such that $\sum_{i=1}^{18}(x_i - \alpha) = 36$ and $\sum_{i=1}^{18}(x_i - \beta)^2 = 90$, where $\alpha$ and $\beta$ are distinct real numbers. If the standard deviation of these observation is 1, then the value of $|\alpha - \beta|$ is ____.

Answer 1

4

Answer 2

1. Relate sum of deviations to the mean: Given $\sum_{i=1}^{18}(x_i - \alpha) = 36$. This expands to $\sum_{i=1}^{18}x_i - 18\alpha = 36$. Let $\bar{x}$ be the mean of the observations, so $\sum_{i=1}^{18}x_i = 18\bar{x}$. Substituting this, we get $18\bar{x} - 18\alpha = 36$. Dividing by 18, we obtain $\bar{x} - \alpha = 2$.

2. Use the variance formula: The standard deviation is given as $\sigma = 1$, so the variance is $\sigma^2 = 1^2 = 1$. The variance is defined as $\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2$. For $n=18$ and $\sigma^2=1$, we have $\sum_{i=1}^{18}(x_i - \bar{x})^2 = n\sigma^2 = 18 \times 1 = 18$.

3. Relate the second given sum to the mean and variance: We use the identity: $\sum_{i=1}^{n}(x_i - c)^2 = \sum_{i=1}^{n}(x_i - \bar{x})^2 + n(\bar{x} - c)^2$. Here, $n=18$, $c=\beta$, $\sum_{i=1}^{18}(x_i - \beta)^2 = 90$, and $\sum_{i=1}^{18}(x_i - \bar{x})^2 = 18$. Substituting these values: $90 = 18 + 18(\bar{x} - \beta)^2$ $90 - 18 = 18(\bar{x} - \beta)^2$ $72 = 18(\bar{x} - \beta)^2$ $(\bar{x} - \beta)^2 = \frac{72}{18} = 4$ Taking the square root, we get $|\bar{x} - \beta| = 2$.

4. Solve for $|\alpha - \beta|$: We have two equations: (a) $\bar{x} - \alpha = 2$ (b) $|\bar{x} - \beta| = 2$, which implies $\bar{x} - \beta = 2$ or $\bar{x} - \beta = -2$.

   Case 1: $\bar{x} - \alpha = 2$ and $\bar{x} - \beta = 2$. Subtracting these equations: $(\bar{x} - \alpha) - (\bar{x} - \beta) = 2 - 2$, which simplifies to $\beta - \alpha = 0$, or $\alpha = \beta$. This contradicts the given condition that $\alpha$ and $\beta$ are distinct real numbers.

   Case 2: $\bar{x} - \alpha = 2$ and $\bar{x} - \beta = -2$. Subtracting these equations: $(\bar{x} - \alpha) - (\bar{x} - \beta) = 2 - (-2)$, which simplifies to $\beta - \alpha = 4$. Therefore, $|\alpha - \beta| = |-(\beta - \alpha)| = |-4| = 4$.

The value of $|\alpha - \beta|$ is 4.